YEAR 10 PROGRAM TERM 1 1. Revision of number operations 3 + T wk 2 2. Expansion 3 + T wk 4 3. Factorisation 7 + T wk 6 4. Algebraic Fractions 4 + T wk 7 5. Formulae 5 + T wk 9 6. Linear Equations 10 +T wk 11 7. Measurement 5 + T wk 1 TERM 2 8. Pythagoras 6 + T wk 2 9. Right Angled Trigonometry 9 + T wk 4 EXAM 1 (Modules 1-10) Week 6 10. Congruence and Similarity 7 + T wk 8 11. Quadratic Equations 10 + T wk 1 TERM 3 12. Co-ordinate Geometry 8 + T wk 3 13. Simultaneous Linear Equations 7 + T wk 5 14. Quadratic Functions 11 + T wk 8 15. Relations and Functions 6 + T wk10 16. Exponential Functions 8 + T wk 1 TERM 4 17. Conic Sections 4 + T wk 2 18. Finance 7 + T wk 3 19. Probability 9 + T wk 5 20. Statistics 4 + T wk 7 21. Bivariate Statistics 6 + T wk 8 EXAM 2 (Modules 10-25)
Year 10 proficiency strands At this year level: Understanding includes applying the four operations to algebraic fractions, finding unknowns in formulas after substitution, making the connection between equations of relations and their graphs, comparing simple and compound interest in financial contexts and determining probabilities of two and three step experiments. Fluency includes factorising and expanding algebraic expressions, using a range of strategies to solve equations and using calculations to investigate the shape of data sets. Problem solving includes calculating the surface area and volume of a diverse range of prisms to solve practical problems, finding unknown lengths and angles using applications of trigonometry, using algebraic and graphical techniques to find solutions to simultaneous equations and inequalities, and investigating independence of events. Reasoning includes formulating geometric proofs involving congruence and similarity, interpreting and evaluating media statements and interpreting and comparing data sets. Year 10 achievement standard By the end of Year 10, students recognise the connection between simple and compound interest. They solve problems involving linear equations and inequalities. They make the connections between algebraic and graphical representations of relations. Students solve surface area and volume problems relating to composite solids. They recognise the relationships between parallel and perpendicular lines. Students apply deductive reasoning to proofs and numerical exercises involving plane shapes. They compare data sets by referring to the shapes of the various data displays. They describe bivariate data where the independent variable is time. Students describe statistical relationships between two continuous variables. They evaluate statistical reports. Students expand binomial expressions and factorise monic quadratic expressions. They find unknown values after substitution into formulas. They perform the four operations with simple algebraic fractions. Students solve simple quadratic equations and pairs of simultaneous equations. They use triangle and angle properties to prove congruence and similarity. Students use trigonometry to calculate unknown angles in right-angled triangles. Students list outcomes for multi-step chance experiments and assign probabilities for these experiments. They calculate quartiles and inter-quartile ranges.
TERM 1 1. Indices 3 + T Apply index laws to numerical expressions with integer indices. (9 ACMNA209) simplifying and evaluating numerical expressions, using both positive and negative integer indices. Extend and apply the index laws to variables, using positive integer indices and the zero index. (9 ACMNA212) understanding that index laws apply to variables as well as numbering Express numbers in scientific notation. (9 ACMNA210) representing extremely large and small numbers in scientific notation, and expressing numbers in scientific notation as whole numbers or decimals. Simplify algebraic products and quotients using index laws (10 ACMNA231) applying knowledge of index laws to algebraic terms, and simplifying algebraic expressions using both positive and negative integral indices Define rational and irrational numbers and perform operations with surds and fractional indices. (10A ACMNA264) understanding that the real number system includes irrational numbers extending the index laws to rational number indices performing the four operations with surds A Index laws m n m n a a a a a m n a mn n a a b b 0 a 1 n n 1 n n a a a m n m m 1 n a a a m n n m m m a n m n a a n 1 n a n n n ab a b 1 n n a a B C Rational (fractional) indices Scientific notation
2. Expansion 3 + T Apply the distributive law to the expansion of algebraic expressions, including binomials and collect like terms where appropriate. (9 ACMNA213) understanding that the distributive law can be applied to algebraic expressions as well as numbers Expand binomial products and factorise monic quadratic expressions using a variety of strategies. (10 ACMNA233) exploring the method of completing the square to factorise quadratic expressions and solve quadratic equations identifying and using common factors, including binomial expressions, to factorise algebraic expressions using the technique of grouping in pairs using the identities for perfect squares and the difference of squares to factorise quadratic expressions A Revision of expansions laws and Further expansion B The binomial expansion
3. Factorisation 7 + T Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts. (10A ACMNA269) writing quadratic equations that represent practical problems A Revision of factorisation B Factorising expressions with four terms C Factorising quadratic trinomials D Miscellaneous factorisation
4. Algebraic Fractions 4 + T Apply the four operations to simple algebraic fractions with numerical denominators (10 ACMNA232) expressing the sum and difference of algebraic fractions with a common denominator using the index laws to simplify products and quotients of algebraic fractions A Evaluating algebraic fractions B Simplifying algebraic fractions C Adding and subtracting algebraic fractions D Multiplying and dividing algebraic fractions
5. Formulae 5 + T Substitute values into formulas to determine an unknown (10 ACMNA234) solving simple equations arising from formulas A Formula construction B Formula substitution C Formula rearrangement D Rearrangement and substitution E Formulae by induction
6. Linear Equations and Inequalities 10 + T Solve problems involving linear equations, including those derived from formula (10 ACMNA235) representing word problems with simple linear equations and solving them to answer questions Solve linear inequalities and graph their solutions on a number line (10 ACMNA236) solving simple linear inequalities graphing the solution set of an inequality on a number line Solve linear equations involving simple algebraic fractions (10 ACMNA240) solving a wide range of linear equations, including those involving one or two simple algebraic fractions, and checking solutions by substitution representing word problems, including those involving fractions, as equations and solving them to answer the question A Solving linear equations B Fractional equations C Problem solving with linear equations D Linear inequalities E Solving linear inequalities F Problem solving with linear inequalities
7. Measurement 5 + T Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites (8 ACMMG196) establishing and using formulas for areas such as trapeziums, rhombuses and kites Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving circumference and area (8 ACMMG197) investigating the circumference and area of circles with materials or by measuring, to establish an understanding of formulas Investigating the area of circles using a square grid or by rearranging a circle divided into sectors Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume (8 ACMMG198) investigating the relationship between volumes of rectangular and triangular prisms Calculate the surface area and volume of cylinders and solve related problems (9 ACMMG217) analysing nets of cylinders to establish formulas for surface area connecting the volume and capacity of a cylinder to solve authentic problems Solve problems involving the surface area and volume of right prisms (9 ACMMG218) solving practical problems involving surface area and volume of right prisms Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids. (10 ACMMG242) investigating and determining the volumes and surface areas of composite solids by considering the individual solids from which they are constructed Solve problems involving surface area and volume of right pyramids, right cones, spheres and related composite solids. (10A ACMMG271) using formulas to solve problems using authentic situations to apply knowledge and understanding of surface area and volume A Unit conversions B Perimeter and Area of 2D shapes C Rectangles Circles Surface area and volume of 3D shapes Prisms and Cylinders
Pyramids and Cones Spheres
TERM 2 8. Pythagoras Theorem 6 + T Investigate Pythagoras Theorem and its application to solving simple problems involving right-angled triangles (9 ACMMG222) understanding that Pythagoras Theorem is a useful tool in determining unknown lengths in right-angled triangles and has widespread applications recognising that right-angled triangle calculations may generate results that can be integers, fractions or irrational numbers Pythagoras Theorem and trigonometry to solving three-dimensional problems in rightangled triangles (10A ACMMG276) investigating the applications of Pythagoras Theorem in authentic problems A Pythagoras theorem B The converse of Pythagoras theorem C Pythagorean triples D Problem solving using Pythagoras theorem E Three-dimensional problems
9. Right Angled Trigonometry 9 + T Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles (9 ACMMG223) developing understanding of the relationship between the corresponding sides of similar right-angled triangles Apply trigonometry to solve right-angled triangle problems (9 ACMMG224) understanding the terms adjacent and opposite sides in a right-angled triangle selecting and accurately using the correct trigonometric ratio to find unknown sides (adjacent, opposite and hypotenuse) and angles in right-angled triangles Solve right-angled triangle problems including those involving direction and angles of elevation and depression (9 ACMMG245) applying Pythagoras Theorem and trigonometry to problems in surveying and design Pythagoras Theorem and trigonometry to solving three-dimensional problems in rightangled triangles (10A ACMMG276) investigating the applications of Pythagoras Theorem in authentic problems A Labelling sides of a right angled triangle B Trigonometric ratios C Finding side lengths D Finding angles E Problem solving using trigonometry F Bearings G 3-dimensional problem solving
10. Congruence and Similarity 7 + T Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (9 ACMMG220) establishing the conditions for similarity of two triangles and comparing this to the conditions for congruence using the properties of similarity and ratio, and correct mathematical notation and language, to solve problems involving enlargement (for example, scale diagrams) using the enlargement transformation to establish similarity understanding that similarity and congruence help describe relationships between geometrical shapes and are important elements of reasoning and proof Solve problems using ratio and scale factors in similar figures (10 ACMMG221) establishing the relationship between areas of similar figures and the ratio of corresponding sides (scale factor) Formulate proofs involving congruent triangles and angle properties (10 ACMMG243) applying an understanding of relationships to deduce properties of geometric figures (for example the base angles of an isosceles triangle are equal) Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes (10 ACMMG244) distinguishing between a practical demonstration and a proof (for example demonstrating triangles are congruent by placing them on top of each other, as compared to using congruence tests to establish that triangles are congruent) performing a sequence of steps to determine an unknown angle giving a justification in moving from one step to the next communicating a proof using a sequence of logically connected statements A Congruence of figures B Congruent triangles C Proof using congruence D Similarity E Similar triangles F Areas and volumes of similar figures
11. Quadratic Equations 10 + T Solve simple quadratic equations using a range of strategies (10 ACMNA241) using a variety of techniques to solve quadratic equations, including grouping, completing the square, the quadratic formula and choosing two integers with the required product and sum Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts. (10A ACMNA269) writing quadratic equations that represent practical problems A Quadratic equations of the form x B The Null Factor law C Solution by factorisaton D Completing the square E The quadratic formula F Problem solving 2 k
TERM 3 12. Coordinate Geometry 8 + T Find the distance between two points located on a Cartesian plane using a range of strategies, including graphing software (9 ACMNA214) investigating graphical and algebraic techniques for finding distance between two points using Pythagoras theorem to calculate distance between two points Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (9 ACMNA294) investigating graphical and algebraic techniques for finding midpoint and gradient recognising that the gradient of a line is the same as the gradient of any line segment on that line Sketch linear graphs using the coordinates of two points and solve linear equations (9 ACMNA215) determining linear rules from suitable diagrams, tables of values and graphs and describing them using both words and algebra Solve problems involving parallel and perpendicular lines (10 ACMNA238) solving problems using the fact that parallel lines have the same gradient and conversely that if two lines have the same gradient then they are parallel solving problems using the fact that the product of the gradients of perpendicular lines is -1 and conversely that if the product of the gradients of two lines is -1 then they are perpendicular A Distance between two points B Midpoints C Gradient D Parallel and perpendicular lines E Equations of straight lines F Graphing lines from equations G Finding the equation of a line
13. Simultaneous Linear Equations 7 + T Solve linear simultaneous equations, using algebraic and graphical techniques including using digital technology (10 ACMNA237) associating the solution of simultaneous equations with the coordinates of the intersection of their corresponding graphs A Graphical solution B Solution by substitution C Solution by elimination D Problem solving
14. Quadratic Functions 11 + T Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (9 ACMNA296) graphing parabolas, and circles connecting x -intercepts of a graph to a related equation Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate (10 ACMNA239) sketching graphs of parabolas, and circles applying translations, reflections and stretches to parabolas and circles sketching the graphs of exponential functions using transformations Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (10A ACMNA267) applying transformations, including translations, reflections in the axes and stretches to help graph parabolas, rectangular hyperbolas, circles and exponential functions A Quadratic functions B Graphs of quadratic functions C Axes intercepts D Axis of symmetry E Vertex
15. Relations and Functions 6 + T Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate (10 ACMNA239) sketching graphs of parabolas, and circles applying translations, reflections and stretches to parabolas and circles sketching the graphs of exponential functions using transformations A Relations B Functions C Function notation D Transforming y f x
16. Exponential Functions and Logarithms 8 + T Solve simple exponential equations (10A ACMNA270) investigating exponential equations derived from authentic mathematical models based on population growth Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (10A ACMNA267) applying transformations, including translations, reflections in the axes and stretches to help graph parabolas, rectangular hyperbolas, circles and exponential functions Use the definition of a logarithm to establish and apply the laws of logarithms (10A ACMNA265) investigating the relationship between exponential and logarithmic expressions simplifying expressions using the logarithm laws A Exponential functions B Graphs of exponential functions C Growth and decay D Exponential equations E Logarithms
TERM 4 17. Conic Sections 4 + T Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (9 ACMNA296) graphing parabolas, and circles connecting x -intercepts of a graph to a related equation Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate (10 ACMNA239) sketching graphs of parabolas, and circles applying translations, reflections and stretches to parabolas and circles sketching the graphs of exponential functions using transformations Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (10A ACMNA267) applying transformations, including translations, reflections in the axes and stretches to help graph parabolas, rectangular hyperbolas, circles and exponential functions A Circles B Ellipses C Hyperbolae
18. Finance 7 + T Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (8 ACMNA187) using percentages to solve problems, including those involving mark-ups, discounts, and GST Solve problems involving profit and loss, with and without digital technologies (8 ACMNA189) expressing profit and loss as a percentage of cost or selling price, comparing the difference investigating the methods used in retail stores to express discounts Solve problems involving simple interest (9 ACMNA211) understanding that financial decisions can be assisted by mathematical calculations Connect the compound interest formula to repeated applications of simple interest using appropriate digital technologies (10 ACMNA229) working with authentic information, data and interest rates to calculate compound interest and solve related problems A Business calculations B Appreciation and depreciation C Simple interest D Compound interest
19. Probability 9 + T List the outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (9 ACMSP225) conducting two-step chance experiments using systematic methods to list outcomes of experiments and to list outcomes favourable to an event comparing experiments which differ only by being undertaken with replacement or without replacement Calculate relative frequencies from given or collected data to estimate probabilities of events involving and or or (9 ACMSP225) using Venn diagrams or two-way tables to calculate relative frequencies of events involving and, or questions using relative frequencies to find an estimate of probabilities of and, or events Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence (10 ACMSP246) recognising that an event can be dependent on another event and that this will affect the way its probability is calculated Using the language of if then, given, of, knowing that to investigate conditional statements and identify common mistakes in interpreting such language (10 ACMSP247) using two-way tables and Venn diagrams to understand conditional statements using arrays and tree diagrams to determine probabilities A Theoretical probability B Compound events C Expectation D Conditional probability
20. Statistics 4 + T Determine quartiles and interquartile range (10 ACMSP248) finding the five-number summary (minimum and maximum values, median and upper and lower quartiles) and using its graphical representation, the box plot, as tools for both numerically and visually comparing the centre and spread of data sets Construct and interpret box plots and use them to compare data sets (10 ACMSP249) understanding that box plots are an efficient and common way of representing and summarising data and can facilitate comparisons between data sets using parallel box plots to compare data about the age distribution of Aboriginal and Torres Strait Islander people with that of the Australian population as a whole Compare shapes of box plots to corresponding histograms and dot plots (10 ACMSP250) investigating data in different ways to make comparisons and draw conclusions Calculate and interpret the mean and standard deviation of data and use these to compare data sets (10 ACMSP278) using the standard deviation to describe the spread of a set of data using the mean and standard deviation to compare numerical data sets Evaluate statistical reports in digital media and elsewhere for information on their planning and implementation (10A ACMSP277) evaluating the appropriateness of sampling methods in reports where statements about a population are based on a sample evaluating whether graphs in a report could mislead, and whether graphs and numerical information support the claims A Discrete data B Continuous data C Measuring the centre D Cumulative data E Measuring the spread F Box plots G Standard deviation H Evaluating reports
21. Bivariate Statistics 6 + T Use scatter plots to investigate and comment on relationships between two numerical variables (10 ACMSP251) using authentic data to construct scatter plots, make comparisons and draw conclusions Investigate and describe bivariate numerical data where the independent variable is time (10 ACMSP252) investigating biodiversity changes in Australia since European occupation constructing and interpreting data displays representing bivariate data over time Use information technologies to investigate bivariate numerical data sets. Where appropriate use a straight line to describe the relationship allowing for variation (10A ACMSP279) investigating different techniques for finding a line of best fit A Line graphs B Scatter plots C Correlation D Measuring correlation E Line of best fit